2.2 Övningar

(Skillnad mellan versioner)

Versionen från 5 juni 2007 kl. 08.08

Beräkna integralerna

a)	$\displaystyle \int_{1}^{2} \displaystyle\frac{dx}{(3x-1)^4}$ genom att använda substitution $u=3x-1$
b)	$\displaystyle \int (x^2+3)^5x \, dx$ genom att använda substitution $u=x^2+3$
c)	$\displaystyle \int x^2 e^{x^3} \, dx$ genom att använda substitution $u=x^3$

Facit

Facit till alla delfrågorna

a)	$\displaystyle\frac{13}{1000}$
b)	$\displaystyle\frac{x^{10}}{12}+C$
c)	$\displaystyle\frac{1}{3}e^{\scriptstyle x^3}+C$

Beräkna integralerna

a)	$\displaystyle\int_{0}^{\pi} \cos 5x\, dx$	b)	$\displaystyle\int_{0}^{1/2} e^{2x+3}\, dx$
c)	$ \displaystyle\int_{0}^{5} \sqrt{3x + 1} \, dx$	d)	$\displaystyle\int_{0}^{1} \sqrt[\scriptstyle3]{1 - x}\, dx$

Facit

Facit till alla delfrågorna

a)	$0$	b)	$\displaystyle\frac{1}{2}(e^4-e^3)$
c)	$14$	d)	$\displaystyle\frac{3}{4}$

Beräkna integralerna

a)	$\displaystyle\int 2x \sin x^2\, dx$	b)	$\displaystyle\int \sin x \cos x\, dx$
c)	$ \displaystyle\int \displaystyle\frac{\ln x}{x}\, dx$	d)	$\displaystyle\int \displaystyle\frac{x+1}{x^2+2x+2}\, dx$
e)	$ \displaystyle\int \displaystyle\frac{x}{x^2+1}\, dx$	f)	$\displaystyle\int \displaystyle\frac{\sin \sqrt{x}}{\sqrt{x}}\, dx$

Facit

Facit till alla delfrågorna

a)	$\cos x^2+C$	b)	$\displaystyle\frac{\cos2x}{4}+C$
c)	$(\ln x)^2$	d)	$\displaystyle\frac{1}{2}\ln\left(x^2+2x+2\right)+C$
e)	$\displaystyle\frac{1}{2}\ln\left(x^2+1\right)+C$	f)	$-2\cos\sqrt{x}+C$

Använd formeln $$\int \frac{dx}{x^2+1} = \arctan x + C$$ för att beräkna integralerna

a)	$\displaystyle\int \displaystyle\frac{dx}{x^2+4}$	b)	$\displaystyle\int \displaystyle\frac{dx}{(x-1)^2+3}$
c)	$ \displaystyle\int e\displaystyle\frac{dx}{x^2+4x+8}$	d)	$\displaystyle\int \displaystyle\frac{x^2}{x^2 +1}\, dx$

 <tr align="left">
 <td class="ntext">a)</td>
-<td class="ntext" width="50%">$\cosx^2+C$</td>
+<td class="ntext" width="50%">$\cos x^2+C$</td>
 <td class="ntext">b)</td>
 <td class="ntext" width="50%">$\displaystyle\frac{\cos2x}{4}+C$</td>